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In the node-weighted prize-collecting Steiner tree problem (NW-PCST) we aregiven an undirected graph $G=(V,E)$, non-negative costs $c(v)$ and penalties$\pi(v)$ for each $v \in V$. The goal is to find a tree $T$ that minimizes thetotal cost of the vertices spanned by $T$ plus the total penalty of verticesnot in $T$. This problem is well-known to be set-cover hard to approximate.Moss and Rabani (STOC'01) presented a primal-dualLagrangean-multiplier-preserving $O(\ln |V|)$-approximation algorithm for thisproblem. We show a serious problem with the algorithm, and present a new,fundamentally different primal-dual method achieving the same performanceguarantee. Our algorithm introduces several novel features to the primal-dualmethod that may be of independent interest.

An LMP O(log n)-Approximation Algorithm for Node Weighted Prize Collecting Steiner Tree